MPI-INF/SWS Research Reports 1991-2021

# MPI-I-95-1-022

## New deterministic algorithms for counting pairs of intersecting segments and off-line triangle range searching

### Pellegrini, M.

#### July 1995, 12 pages.

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##### Status: available - back from printing

We describe a new method for decomposing planar sets of segments and points. Using this method we obtain new efficient {\em deterministic} algorithms for counting pairs of intersecting segments, and for answering off-line triangle range queries. In particular we obtain the following results: \noindent (1) Given $n$ segments in the plane, the number $K$ of pairs of intersecting segments is computed in time $O(n^{1+\epsilon} + K^{1/3}n^{2/3 + \epsilon})$, where $\epsilon >0$ an arbitrarily small constant. \noindent (2) Given $n$ segments in the plane which are coloured with two colours, the number of pairs of {\em bi-chromatic} intersecting segments is computed in time $O(n^{1+\epsilon} + K_m^{1/3}n^{2/3 +\epsilon})$, where $K_m$ is the number of {\em mono-chromatic} intersections, and $\epsilon >0$ is an arbitrarily small constant. \noindent (3) Given $n$ weighted points and $n$ triangles on a plane, the sum of weights of points in each triangle is computed in time $O(n^{1+\epsilon} + {\cal K}^{1/3}n^{2/3 +\epsilon})$, where ${\cal K}$ is the number of vertices in the arrangement of the triangles, and $\epsilon>0$ an arbitrarily small constant. The above bounds depend sub-linearly on the number of intersections among segments $K$ (resp. $K_m$, ${\cal K}$), which is desirable since $K$ (resp. $K_m$, ${\cal K}$) can range from zero to $O(n^2)$. All of the above algorithms use optimal $\Theta(n)$ storage. The constants of proportionality in the big-Oh notation increase as $\epsilon$ decreases. These results are based on properties of the sparse nets introduced by Chazelle.

URL to this document: https://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1995-1-022

BibTeX
@TECHREPORT{Pellegrini95,
AUTHOR = {Pellegrini, M.},
TITLE = {New deterministic algorithms for counting pairs of intersecting segments and off-line triangle range searching},
TYPE = {Research Report},
INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},